Controlled synchronization of three co-rotating exciters based on a circular distribution in a vibratory system

In this article, an engineering problem of three co-rotating exciters with the circular distribution in a vibrating system is investigated. The dynamical model constructed by the motion differential equations is established. By introducing the small parameter averaged method in the dynamic equation, the synchronization and stability conditions of the electromechanical coupling dynamical model is derived. To illustrate the necessity of the controlling method, the self-synchronization of the vibrating system is firstly analyzed with the theory, numerical simulations and experiments. With the self-synchronization results, it is indicated that the ellipse trajectory which is needed in the industry can’t be realizefd by the self-synchronization motion of the vibrating system. And then, a fuzzy PID controlling method based on the master–slave controlling strategy is introduced in the vibrating system to realize the controlled synchronization. The Lyapunov stability criterion is given to certify the stability of the controlling system. Through some simulations and experiments, the effectiveness of controlled synchronization is illustrated in the discussion. Finally, the present work illuminates the feasibility and practicality for designing some new types of vibrating screens in the industry.


Synchronization and stability analysis of dynamical differential equation The establishment of the theoretical model
The model of the vibratory screen can be simplified as the theoretical model of a vibrating system shown in Fig. 1.And the significance of mathematical symbols in this paper are all listed in Table 1.The vibrating system is constructed from top to the bottom.Two inductor motors with ERs are symmetrically installed on a rigid frame along the vertical axis.One motor is installed under the rigid frame on the vertical axis.o i (i = 1,2,3) are respectively the rotating center of three ERs.The frame is supported by four springs which provide the stiffness in the x and y directions.o is the center of the rigid frame.M is the total mass of the vibrating system, M = m + m i (i = 1, 2, 3) .m is the mass of the frame and oo i = l i (i = 1, 2, 3) .J and J p are respectively the inertia moment of the vibrating system and the rigid frame, J = Ml 2 e ≈ J p + 3 i=1 m i (l 2 i + r 2 ) .l e is equivalent rotation radius.f x , f y and f ψ are the damping coefficients, f ψ = l 2 0 (f x sin 2 β + f y cos 2 β) .k x , k y and k ψ are the stiffnesses, k ψ = l 2 0 (k x sin 2 β + k y cos 2 β) .f i (i = 1,2,3) respectively represents damping coefficients of the inductor motor.The parameters of the vibrating system are given in Table 2. www.nature.com/scientificreports/According to Fig. 1, the mathematical model of the vibrating system can be derived with Lagrange equation.
Equation 1 contains four variables, the first is the kinetic energy T which is composed of three parts.They are respectively the item of translational energy of the rigid body T teb = m(ẋ 2 + ẏ2 )/2 , rotational energy of the rigid body T reb = J b ψ2 /2 and the kinetic energy of three ERs T kee = , where, According to the transformation of the coordinate system, it can be derived as δ 0 = x y , χ = cos ψ − sin ψ sin ψ cos ψ , δ ′′ i = l i cos θ i + r cos ϕ i l i sin θ i + r sin ϕ i .Thus, the total kinetic energy of the system can be expressed The nomenclature table of the symbols.

Symbol Significance m i
The mass of each ERs

J p
The moment of inertia of the rigid frame

J
The moment of inertia of the vibrating system The stiffness coefficients of the vibration system in the x, y and ψ directions r The eccentric radius of the inductor motors

M
The total mass of the vibration system The damping coefficients of the vibration system in the x, y and ψ directions The distance between the center of the body and the rotating center of motors

J i
The moment of inertia of the inductor motor d-, q-The d-and q-axes in rotor field-oriented coordinate www.nature.com/scientificreports/as T = T teb + T reb + T kee .The second is the potential energy V which can be expressed as V = (k x x 2 + k y y 2 + k ψ ψ 2 )/2 .T h e t h i rd i s t h e e n e r g y d i s s ip at i on D d e s c r i b e d a s T h e l a s t i t e m s a r e t h e g e n e r a l i z e d f o r c e (0, 0, 0, T e1 , T e2 , T e3 ) T and the generalized coordinate q = (x, y, ψ, ϕ 1 , ϕ 2 , ϕ 3 ) T .Through Eq. (1), the differential equations of the electromechanical coupling dynamical model can be expressed as Eq. ( 2) with the Lagrange function.
where, T Li is the torque loads of three motors and it can be derived as Eq.(3).
The item T e in Eq. ( 2) is the electromagnetic torque of the inductor motor.Thus, the model of the inductor motor should be established.In this article, the category of the inductor motor is the squirrel-cage motor.According to the feature of this kind of motor, its rotor winding is short circuit, which can be expressed as the mathematical form, u rd = u rq .When the motor is at a stable state, φ rd = constant and φ rq = 0 .With the vari- ables ω-i s -φ r , the model of inductor motor in the d-q rotating coordinate system can be represented as Eq. ( 4) by literature 24 .
The symbols in Eq. ( 4) are all given in Table 1.L ks and R ks respectively represent the leakage inductance and the equivalent resistance of the stator which can be derived as L ks = L s − L 2 m /L r and R ks = R s + L 2 m R r /L 2 r .θ and ω s can respectively be represented with the mathematical formulation θ = (ω + ω s )dt and ω s = L m i sq /φ rd T r .To sustain the stability of the motor speed, the rotor flux-oriented control (RFOC) is introduced in the system which is shown in Fig. 2. In the meanwhile, the parameters of three motors are shown in Table 3.

The analysis of synchronization and stability of the vibrating system
According to the non-linear dynamical theory, the phase difference between motor 1 and 2 can be expressed as pϕ 1 − qϕ 2 = 2α 1 , and the phase difference between motor 2 and 3 can be expressed as qϕ 2 − sϕ 3 = 2α 2 .Set the average phase of three ERs as ϕ which can be represented as ϕ = (ϕ 1 + ϕ 2 + ϕ 3 )/3 .And then the average speed ω 0 in the period T can be derived as ω 0 = T 0 φdt/T .With the small parameter method, the speed and accelerated (2) www.nature.com/scientificreports/speed of three ERs are respectively φi = (1+ε i )ω 0 and φi = εi ω 0 , i = 1,2,3.ε i is the small perturbation parameter.ω 0 that is the mean value of the average speed of three motors equals to a constant.Then, the responses in three directions can be derived as Eq. ( 5) where, ω 2 From the equations above, the feature of the parameters r li , r m , and l i are analyzed in the situation l 1 = l 2 = l 3 .And then, the relationships among the three parameters are illustrated with different parameter η i in Fig. 3.The results represent that the closer the masses of three ERs are, the better their synchronous capability is.
Taking the small parameters into Eq.( 2), Eq. ( 6) can be obtained with the integrate method in the 2π period.
where, the torque loads in Eq. ( 6) can be expressed with small parameters as (5)  The items a ij , b ij and κ i which are the second stability coefficient in Eq. ( 7) are all given in Appendix A. Because a ij and b ij are both associated with r li , Fig. 4 shows their relationship feature with different parameter η i .The results indicates that the parameters a ij and b ij are changed with the various parameter η i and the coupling dynamical feature of the vibrating system demonstrates the best stable capability in the situation η 1 = η 2 = η 3 .By introducing in the small parameter ε 4 and ε 5 , expand the phase differences with the Taylor formula separately and omit the higher order items, α 1 = α 1 + ε 4 and α 2 = α 2 + ε 5 can be obtained.In the meanwhile, the non- dimensional coupling equations which is numbered as ( 8) can be derived as where, A is a matrix of the moment of inertia and B is a stiffness matrix.The symbols in Eq. ( 8) are listed as below.
The electromagnetic torque in Eq. ( 6) can be expressed as where, ) and k e0i can be derived from literature 25 .T e0i is painted as Fig. 5.
where, T eNi (i = 1, 2, 3) are respectively the rated electromagnetic torque of three motors.When the vibrating system satisfies the synchronization criterion, υ = 0 .Equation ( 8) can be simplified as Because matrix A is a non-singular matrix, if the determinant |A| of matrix A dose not equal to zero, the matrix A is an invertible matrix.Thus, Eq. ( 11) can be expressed as where, D = A −1 B .The characteristic equation in Eq. ( 12) can be obtained by det( I − D) = 0. www.nature.com/scientificreports/d j j = 1, 2, 3, 4, 5 are the coefficient items while represents the eigenvalue in Eq. ( 13).The parameters d j (j = 1, 2, 3, 4, 5) are given in Appendix C. When the characteristic equation satisfies the Hurwitz conditions in Eq. ( 14), all the real roots of eigenvalues exist on the left of coordinate (which means to be the negative real roots), the synchronous state of the vibrating system is stable.Otherwise, is unstable. (

Controller design of the controlling system
In this controlling scheme, a master-slave controlling strategy is introduced in the controlling system as shown in Fig. 6. ω t as an input target speed is given into the master motor which is named motor 1.The output speed of motor 1 is divided into three functions.One is feedback to the initial speed to promote the accuracy of the motor speed.Another is as an input variable to realize the controlling strategy by the adaptive fuzzy PID method.In the meanwhile, it is converted to the speeds of motor 2 and 3.The other is changed to the phase through an integrate method and then transferred to the dynamical model.The processes of motor 2 and 3 are similar with motor 1's.
The fuzzy logic model is consisted of two input variables and three output variables.The two input variables are respectively the error e and the change rate of error ec.The three output variables are respectively the proportionality coefficient k p , integral coefficient k I and differential coefficient k D .According to the usual fuzzy rules table, the output variables can be obtained through Eqs.(15) with forty-nine rules which are established in this section.( 14) www.nature.com/scientificreports/

Theoretical analysis of the controlling system
Because the controlling method is applied in this article, the stability of the controlling method should be analyzed.Choosing the speed as the state variable, it can be expressed as ω = φ .And then, Eq. ( 2) can be derived as K T i u i is introduced in the dynamical equation to replace the electromagnetic torque.Where, K T i is represented as the constant of the electromagnetic torque, K Ti = L mi φ rdi /L ri , while u i is considered as the controlling variable i * qsi .W i replaces the load torque which is the uncertain loads in this section, W i = T Li .Set e as the speed error of the motor and then it can be expressed as the difference between the target speed and the actual speed in Eq. ( 17).
Thus, the tracing error E can be given as a column vector, E = [e, ė] T .Because u i is the controlling variable, the controlling law is defined as . Where, ξ(x) is a fuzzy vector.Then the adaptive law of the fuzzy system is represented as γ is a positive constant and P is a positive definite matrix.Taking Eq. ( 18) into Eq.( 16), the dynamical equation with closed loop of the fuzzy controlling system can be expressed as where To guarantee the boundedness of the weight coefficient θ f , the optimal weight coefficient θ * f is introduced in the controlling system with a convex set f which contains θ f , θ f ∈ � f .Thus, θ * f is constructed as Taking Eq. ( 19) and ( 21) into Eq.( 20), the closed dynamical equation of the fuzzy system can be derived as is defined as the minimum approximation error and is expressed as To accord the adaptive law with the condition of the fuzzy system, the difference between the tracing error E and the parameter error θ f − θ * f should be minimum.So, a Lyapunov function is defined as γ is a positive constant.Through introducing a positive definite matrix Q with second order, the Lyapunov equation should be satisfied with matrix P.
To certify the stability of Lyapunov function, the Lyapunov function in Eq. ( 23) is divided into two parts.One is According to the Lyapunov criterion, the derivation of V 1 and V 2 should be obtained, which are respectively Thus, the derivate of Eq. ( 23) can be expressed as Because of −E T QE/2 ≤ 0 , only if the appropriate parameter is chosen, V ≤ 0 can be obtained.According to LaSalle invariance principle, the controlling system is asymptotic stable.The stability of other motors' speed error and the stability of phase error can be acquired with the method above. (15)

Results analysis and discussions
In this section, numerical simulation results corresponding to the dynamical model of Fig. 1 are illustrated.And then, some experiment results are given to verify the simulation results.The feature of vibrating system is discussed.

Numerical simulation of self-synchronization and controlled synchronization
In Fig. 7, the simulation result of the self-synchronization with three ERs are presented.Figure 7a shows the speed of three motors.With the method of constant voltage frequency ratio, the speeds of three motors fluctuate around 60.02 rad/s which can be realized to reach the synchronous speeds.However, the phase differences between motor 1 and 2 with motor 2 and 3 are separately about 108.5° and 119.5° in Fig. 7b.The result illustrates that the vibrating system obviously can't reach the synchronous state with zero phase difference which is needed in the engineering.From Fig. 7c, it can be known that the amplitude of the vibrating system is counteracted due to the phase differences in Fig. 7b.Thus, the vibrating system appears an angle of oscillation with 0.02° in Fig. 7d.From the results in Fig. 7, it can be concluded that the vibrating system can't realize the synchronous state with zero phase difference.And then, the trajectory of the rigid body can't meet the engineering acquirements.
To solve this problem, the fuzzy PID method is introduced in the self-synchronization motion to realize the synchronous state with zero phase difference.From Fig. 8a, the speeds of three motors are all about 60 rad/s which equal to the target speed.Figure 8b demonstrates that the phase differences between motor 1 and 2 with motor 1 and 3 are both approximate to 0. And this result represents that the three ERs realize the controlled synchronization motion.In Fig. 8c, values of the torque load are between 0 and 0.12.These values are less than the absolute values of the electromagnetic torque.Thus, the three motor can operate normally and won't appear the phenomenon of motor blocking.From Fig. 8d,f, the results illustrate that the amplitudes of ERs are superimposed and the angle of swing of the vibrating system is very small.The vibrating system realizes the controlled synchronization motion and the elliptical trace which is needed in the engineering.To expound the feature of distance between motor 1 and 2, the distance is expanded from center to the sides which is shown in Fig. 9.According to Fig. 9, the speeds and phase differences are both similar with them in Fig. 8.However, the torque loads of three motors enlarge in Fig. 9c due to the increasing of the swing angle in Fig. 9e.The results indicate that although the parameters l 1 and l 2 increase with increasing the distance between motor 1 and 2, this change has little influence on the controlled synchronization motion.Because the stable controlling method changes the intrinsic dynamical feature of the vibrating system.Thus, to realize the miniaturization of the vibrating system, the parameters in Fig. 8 are more suitable for the engineering.This is the reason that the vibrating system is in a circular distribution rather than one straight line.

Experimental verification of self-synchronization and controlled synchronization
To certify the correctness of the proposed theory and the consistency with numerical simulation, some experiments of self-synchronization and controlled synchronization are given.The experimental facilities are firstly listed in Fig. 10 which illuminates the experimental procedure in the meanwhile.In this experiment, the frequencies of three inductor motors are respectively set as 35 Hz through three convertors.Through the calculation conversion, the speeds of three motors can be calculated as 73.2 rad/s which can be recognized as the theorical value.In Fig. 11a, the speeds of three motors all have a large fluctuation around 73 rad/s.Due to the existence of the error in the experiment, this result can be recognized to realize the synchronous speed.In Fig. 11b, the phase difference between motor 1 and 2 is about 127°, and this result is similar with the result in Fig. 7b.The same result can be obtained from the phase difference between motor 1 and 3. From Fig. 11c,e, the results indicates that the amplitude responses of three directions are all counteracted.Thus, the experimental result is consistence with the numerical simulation result of the self-synchronization. Figures 12 and 13 both represent the experimental results of the controlled synchronization motion and are respectively responses to the simulation results in Figs. 8 and 9.With the controlling method, the speeds of three motors exist less fluctuation.The phase differences can be considered small enough to realize the controlled synchronization motion.As shown from (d) to (g) in Figs. 12 and 13, the responses amplitudes in three directions are all in superposition state and the values are approximately the same.This result adequately represents that the vibrating system can realize the elliptic

Conclusions
This article investigates controlled synchronization of three co-rotating exciters based on a circular distribution in a vibratory system.Through the theorical feature analysis, the results indicate that the self-synchronization motion is depended the parameter of η i and l i .When l i can't meet the condition of zero phase difference in the self-synchronization motion, the ellipse motion trace in the vibrating system can't be realized.Therefore, the stability of the vibrating system depends on the controlling method and suitable controlling strategy in the controlled synchronization motion which can realize the ellipse motion trace in the vibrating system.Compared with the former researches, this article adopts a circle distribution in the dynamical model.It can further reduce the structure size of the vibrating system, which is the purpose of using the controlled synchronization instead of the self-synchronization.In the meanwhile, an adaptive fuzzy PID method is introduced.Because this method can be independent to the dynamical model and the parameter PID don't need to be founded with lots of time compared with the former methods.

Figure 1 .
Figure 1.Mathematical model of the vibrating system.

3
Mutual inductance of the stator and rotor L ks Leakage inductance of the stator φ sd The flux linkages of the stator in the d-axis φ sq The flux linkages of the stator in the q-axis φ rd The flux linkages of the rotor in the d-axis φ rq The flux linkages of the rotor in the q-axis R s The stator resistance R r The rotor resistance R ks Equivalent resistance of the stator i sd The current of stator in the d-axis i sq The current of stator in the q-axis i rd The current of rotor in the d-axis i rq The current of rotor in the q-axis ω The mechanical speed ω s The synchronous electric angular speed φsd , φsq , φrd , φrq Derivation ofφ sd ,φ sq ,φ rd ,φ rq σ The leakage factor Tr A rotor time constant u sd The voltage of stator in the d-axis u sq The voltage of stator in the q-axis u rd The voltage of rotor in the d-axis u rq The voltage of rotor in the q-axis n p The number of pole-pairs of the induction motor • * The given values or obtained from the given values ω 1 ,ω 2 ,ω 3 The speeds of three motors ϕ 1 ,ϕ 2 ,ϕ The https://doi.org/10.1038/s41598-024-55680-8

Figure 3 .
Figure 3.The feature analysis of three parameters.

Figure 4 .
Figure 4.The parameters a ij and b ij with the relationship of r l in the situation ω 0 = 80 rad/s.

Figure 13 .
Figure 13.Experiment of controlled synchronization with three ERs, α 0 =0 , η 1 = η 2 = η 3 = 1 (motor 1 and 2 approach both sides of the rigid body).(a) speeds, (b) phase difference between motor 1 and 2, (c) phase differences between motor 1 and 3, (d) response in the x direction, (e) response in the y1 direction, (f) response in the y2 direction, (g) The motion trace of the rigid body

Table 2 .
The parameters of the vibrating system.

Table 3 .
The parameters of three motors.